Calculation of in-band radiances via the line-by-line (LBL) approach is computationally intensive. Not only does it require the calculation of spectral molecular absorption coefficients for the series of pressure and temperature pairs encountered in the specified atmosphere, but it also requires the radiation transport equation (RTE) be solved for each spectral point. Because of the fine structure of molecular absorption, especially at higher altitudes (lower pressures), a very finely spaced spectral grid is required. Spectral step size depends upon scenario and accuracy requirements, but values of between 0.01 and 0.00001 cm−1 is common.
The MODTRAN™ solution to this problem is to use a narrow statistical band model. In this approach, the RTE is solved for a relatively narrow spectral band all at once. A number of physical and statistical approximations are made. The quality of the MODTRAN™ results depend largely upon the validity of these approximations for any specific scenario. Over the past 30 years of MODTRAN™ development, much effort has been focused on fine tuning the model to improve accuracy for the terrestrial atmosphere. An earlier version of MODTRAN is described in U.S. Pat. No. 5,884,226.
The integral RTE defines the line-of-sight (LOS) monochromatic diffuse radiant intensity Iv(Ω0;0) at spectral frequency v along a refracted path in the direction specified by solid angle Ω0:
                                          I            v                    ⁡                      (                                          Ω                0                            ;              0                        )                          =                                            ⅇ                              -                                                      τ                    v                                    ⁡                                      (                    s                    )                                                                        ⁢                                          I                v                            ⁡                              (                                                      Ω                    s                                    ;                  s                                )                                              +                                    ∫              0                                                τ                  v                                ⁡                                  (                  s                  )                                                      ⁢                                          ⅇ                                  -                                                            τ                      v                                        ⁡                                          (                                              s                        ′                                            )                                                                                  ⁢                                                J                  v                                ⁡                                  (                                                            Ω                                              s                        ′                                                              ;                                          s                      ′                                                        )                                            ⁢                                                          ⁢                                                ⅆ                                                            τ                      v                                        ⁡                                          (                                              s                        ′                                            )                                                                      .                                                                        (        1        )            This equation states that the observed spectral radiant intensity at the sensor (s=0) equals the spectral radiant intensity directed towards the sensor a distance s away along the refracted path attenuated by the foreground path transmittance e−τv(s) plus the integral of intervening source radiation Jv(Ωs′;s′) attenuated by its foreground transmittance e−τv(s′). The path spectral transmittances in Eq. (1) are defined in terms of dimensionless optical depths, τv(s). The optical depth terms themselves are path integrals over extinction coefficients, bv(s′) [units of reciprocal length].
                                          τ            v                    ⁡                      (            s            )                          ≡                              ∫            0            s                    ⁢                                                    b                v                            ⁡                              (                                  s                  ′                                )                                      ⁢                                                  ⁢                          ⅆ                              s                ′                                                                        (        2        )            The extinction coefficients consists of absorption bvabs(s) and scattering bvsct(s) components, which are computed as sums over atmospheric molecular and particulate species contributions bv,i(s).
                                                                                          b                  v                                ⁡                                  (                  s                  )                                            ≡                            ⁢                                                                    b                    v                    abs                                    ⁡                                      (                    s                    )                                                  +                                                      b                    v                    sct                                    ⁡                                      (                    s                    )                                                                                                                          ≡                            ⁢                                                ∑                                      i                    ⁢                                                                                  ⁢                    species                                                  ⁢                                                                  ⁢                                  [                                                                                    b                                                  v                          ,                          i                                                abs                                            ⁡                                              (                        s                        )                                                              +                                                                  b                                                  v                          ,                          i                                                sct                                            ⁡                                              (                        s                        )                                                                              ]                                                                                                        ≡                            ⁢                                                ∑                                      i                    ⁢                                                                                  ⁢                    species                                                  ⁢                                                                  ⁢                                                      [                                                                                            κ                                                      v                            ,                            i                                                    abs                                                ⁡                                                  (                          s                          )                                                                    +                                                                        κ                                                      v                            ,                            i                                                    sct                                                ⁡                                                  (                          s                          )                                                                                      ]                                    ⁢                                                            ρ                      i                                        ⁡                                          (                      s                      )                                                                                                                              (        3        )            The individual species absorption and scattering coefficients are themselves the product of a cross-section κv,i(s) [units of area] and a density ρi(s) [number per unit volume].
The atmospheric source term Jv(Ωs;s) consists of three distinct components: local thermal emission, single scattered directly transmitted solar (and/or lunar) irradiance and diffuse radiation scattered into the LOS
                                                                                          J                  v                                ⁡                                  (                                                            Ω                      s                                        ;                    s                                    )                                            =                                                                                          b                      v                      abs                                                                                      b                        v                                            ⁡                                              (                        s                        )                                                                              ⁢                                                            B                      v                                        ⁡                                          (                                              T                        s                                            )                                                                      ⁢                                                                  +                                                                        local              ⁢                                                          ⁢              thermal              ⁢                                                          ⁢              emission                                                                                                                                b                    v                    sct                                                                              b                      v                                        ⁡                                          (                      s                      )                                                                      ⁢                                                      p                    v                                    ⁡                                      (                                                                  Ω                        s                                            ,                                                                        Ω                          s                          sun                                                ;                        s                                                              )                                                  ⁢                                  ⅇ                                      -                                                                  τ                        v                        sun                                            ⁡                                              (                        s                        )                                                                                            ⁢                                  F                  v                  sun                                            ⁢                                                          +                                                          single              ⁢                                                          ⁢              scatter              ⁢                                                          ⁢              solar                                                                                                                                b                    v                    sct                                    ⁡                                      (                    s                    )                                                                                        b                    v                                    ⁡                                      (                    s                    )                                                              ⁢                                                ∫                                      4                    ⁢                    π                                                                                                          ⁢                                                                            p                      v                                        ⁡                                          (                                                                        Ω                          s                                                ,                                                                              Ω                            ′                                                    ;                          s                                                                    )                                                        ⁢                                                            I                      v                                        ⁡                                          (                                                                        Ω                          ′                                                ;                        s                                            )                                                        ⁢                                                                          ⁢                                      ⅆ                                          Ω                      ′                                                                                                                              multiple              ⁢                                                          ⁢                              scatter                .                                                                        (        4        )            The first term, the thermal emission, is the product of the Planck blackbody function Bv(Ts) at local temperature Ts and the local emissivity, which is represented as a ratio of absorption to extinction coefficients. This expression for thermal emission assumes local thermodynamic equilibrium (LTE) conditions—an assumption, valid in the lower terrestrial atmosphere, that molecular collisions occur in sufficient frequency to control upper state population densities.
The latter two terms in Eq. (4) are the path scattered radiance. The single scatter solar radiation is the product of the top-of-atmosphere (TOA) solar irradiance Fvsun, the scattering point to sun transmittance e−τvsun(s), the effective scattering phase function pv(Ωs,Ωssun;s), and the single scattering albedo, equal to the ratio of scattering to extinction coefficients. The multiply scattered radiation is the integral over all directions (4π steradians) of the incoming diffuse radiation scattered into the LOS. The dependence of the multiple scattering on path radiance itself significantly complicates the radiation transport equation, transforming it into a fully coupled problem. MODTRAN™ uses the DISORT model (below), a discrete ordinate algorithm, to solve the coupled equations. The effective phase function that appears in both scattered radiance source terms is a scattering coefficient weighted average of individual species i scattering phase functions pv,i(Ωs,Ωs′;s):
                                          p            v                    ⁡                      (                                          Ω                s                            ,                                                Ω                  s                  ′                                ;                s                                      )                          ≡                                            ∑                              i                ⁢                                                                  ⁢                species                                      ⁢                                                  ⁢                                                            b                                      v                    ,                    i                                    sct                                ⁡                                  (                  s                  )                                            ⁢                                                p                                      v                    ,                    i                                                  ⁡                                  (                                                            Ω                      s                                        ,                                                                  Ω                        s                        ′                                            ;                      s                                                        )                                                                                        b              v              sct                        ⁡                          (              s              )                                                          (        5        )            The phase functions of Eq. (4) are all unit normalized, with dimensions of reciprocal steradians (1/sr). It is somewhat more common to define phase functions as dimensionless with a 4π sr normalization, but MODTRAN™ uses the unit normalization to eliminate all constant factors from the source function equation.
Substituting the source term of Eq. (4) into the integral RTE, the spectral radiant intensity of Eq. (1) can be rewritten as absorption optical depth τvabs(s) and scattering optical depth τvsct(s) integrals:
                                                                        I                v                            ⁡                              (                                                      Ω                    0                                    ;                  0                                )                                      =                                                            ⅇ                                      -                                                                  τ                        v                                            ⁡                                              (                        s                        )                                                                                            ⁢                                                      I                    v                                    ⁡                                      (                                                                  Ω                        s                                            ;                      s                                        )                                                              +                                                ∫                  0                                                            τ                      v                      abs                                        ⁡                                          (                      s                      )                                                                      ⁢                                                      ⅇ                                          -                                                                        τ                          v                                                ⁡                                                  (                                                      s                            ′                                                    )                                                                                                      ⁢                                                            B                      v                                        ⁢                                                                                  (                                          T                                              s                        ′                                                              )                                    ⁢                                      ⅆ                                                                  τ                        v                        abs                                            ⁡                                              (                                                  s                          ′                                                )                                                                                                        +                                                ∫                  0                                                            τ                      v                      sct                                        ⁡                                          (                      s                      )                                                                      ⁢                                                                            ⅇ                                              -                                                                              τ                            v                                                    ⁡                                                      (                                                          s                              ′                                                        )                                                                                                                ⁡                                          [                                                                                                                                                                                                                                        p                                    v                                                                    ⁡                                                                      (                                                                                                                  Ω                                                                                  s                                          ′                                                                                                                    ,                                                                                                                        Ω                                                                                      s                                            ′                                                                                    sun                                                                                ;                                                                                  s                                          ′                                                                                                                                                      )                                                                                                  ⁢                                                                  ⅇ                                                                      -                                                                                                                  τ                                        v                                        sun                                                                            ⁡                                                                              (                                                                                  s                                          ′                                                                                )                                                                                                                                                                            ⁢                                                                  F                                  v                                  sun                                                                                            +                                                                                                                                                                                                                          ∫                                                                  4                                  ⁢                                  π                                                                                                                                                                                              ⁢                                                                                                                                    I                                    v                                                                    ⁡                                                                      (                                                                                                                  Ω                                        ′                                                                            ;                                                                              s                                        ′                                                                                                              )                                                                                                  ⁢                                                                                                      p                                    v                                                                    ⁡                                                                      (                                                                                                                  Ω                                                                                  s                                          ′                                                                                                                    ,                                                                                                                        Ω                                          ′                                                                                ;                                                                                  s                                          ′                                                                                                                                                      )                                                                                                  ⁢                                                                                                                                  ⁢                                                                  ⅆ                                                                      Ω                                    ′                                                                                                                                                                                                                          ]                                                        ⁢                                                                          ⁢                                      ⅆ                                                                  τ                        v                        sct                                            ⁡                                              (                                                  s                          ′                                                )                                                                                                                          ;                ⁢                                  ⁢                                            τ              v              abs                        ⁡                          (              s              )                                ≡                                    ∫              0              s                        ⁢                                                            b                  v                  abs                                ⁡                                  (                                      s                    ′                                    )                                            ⁢                                                          ⁢                              ⅆ                                  s                  ′                                            ⁢                                                          ⁢              and              ⁢                                                          ⁢                                                τ                  v                  sct                                ⁡                                  (                  s                  )                                                              ≡                                    ∫              0              s                        ⁢                                                            b                  v                  sct                                ⁡                                  (                                      s                    ′                                    )                                            ⁢                                                          ⁢                                                ⅆ                                      s                    ′                                                  .                                                                        (        6        )            For band model theory, the key atmospheric attenuation parameter is the transmittance itself, not the optical depth. For that reason, it is convenient to again rewrite the RTE in terms of transmittance, tv(s)=exp [−τv(s)], integrals:
                                          I            v                    ⁡                      (                                          Ω                0                            ;              0                        )                          =                                                            t                v                            ⁡                              (                s                )                                      ⁢                                          I                v                            ⁡                              (                                                      Ω                    s                                    ;                  s                                )                                              +                                    ∫                                                t                  v                  abs                                ⁡                                  (                  s                  )                                            1                        ⁢                                                            t                  v                  sct                                ⁡                                  (                                      s                    ′                                    )                                            ⁢                                                B                  v                                ⁡                                  (                                      T                                          s                      ′                                                        )                                            ⁢                                                          ⁢                              ⅆ                                                      t                    v                    abs                                    ⁡                                      (                                          s                      ′                                        )                                                                                +                                    ∫                                                t                  v                  sct                                ⁡                                  (                  s                  )                                            1                        ⁢                                                                                t                    v                    abs                                    ⁡                                      (                                          s                      ′                                        )                                                  ⁡                                  [                                                                                                                                                                                                        p                                v                                                            ⁡                                                              (                                                                                                      Ω                                                                          s                                      ′                                                                                                        ,                                                                                                            Ω                                                                              s                                        ′                                                                            sun                                                                        ;                                                                          s                                      ′                                                                                                                                      )                                                                                      ⁢                                                                                          t                                v                                sun                                                            ⁡                                                              (                                                                  s                                  ′                                                                )                                                                                      ⁢                                                          F                              v                              sun                                                                                +                                                                                                                                                                                          ∫                                                          4                              ⁢                              π                                                                                                                                                                      ⁢                                                                                                                    I                                v                                                            ⁡                                                              (                                                                                                      Ω                                    ′                                                                    ;                                                                      s                                    ′                                                                                                  )                                                                                      ⁢                                                                                          p                                v                                                            ⁡                                                              (                                                                                                      Ω                                                                          s                                      ′                                                                                                        ,                                                                                                            Ω                                      ′                                                                        ;                                                                          s                                      ′                                                                                                                                      )                                                                                      ⁢                                                                                                                  ⁢                                                          ⅆ                                                              Ω                                ′                                                                                                                                                                                          ]                                            ⁢                                                ⅆ                                                            t                      v                      sct                                        ⁡                                          (                                              s                        ′                                            )                                                                      .                                                                        (        7        )            
Within MODTRAN™, the boundary radiant intensity term Iv(Ωs;s) for the full LOS path is typically zero unless the path terminates at the spherical Earth surface. There is an option to include grey body emission with no reflected source at the path final altitude. For a down-looking sensor whose LOS intersects the ground, the surface radiance includes both emissive and reflective components:
                                                                                          I                  v                                ⁡                                  (                                                            Ω                      s                                        ;                    s                                    )                                            =                                                                                          ɛ                      v                                        ⁡                                          (                                                                        Ω                          s                                                ;                        s                                            )                                                        ⁢                                                            B                      v                                        ⁡                                          (                                              T                        g                                            )                                                                      +                                                                        Surface              ⁢                                                          ⁢              emission                                                                                                            μ                  s                  sun                                ⁢                                                      f                    v                                    ⁡                                      (                                                                  Ω                        s                                            ,                                                                        Ω                          s                          sun                                                ;                        s                                                              )                                                  ⁢                                                      t                    v                    sun                                    ⁡                                      (                    s                    )                                                  ⁢                                  F                  v                  sun                                            ⁢                                                          +                                                                                                                Reflected                    ⁢                                                                                  ⁢                    directly                                                                                                                                          transmitted                      ⁢                                                                                          ⁢                      solar                                        ⁢                                                                                                                                                                                                                                                ⁢                    irradiance                                                                                                                                          ∫                                  2                  ⁢                  π                                                                                              ⁢                                                                    I                    v                                    ⁡                                      (                                                                  Ω                        ′                                            ;                      s                                        )                                                  ⁢                                                      f                    v                                    ⁡                                      (                                                                  Ω                        s                                            ,                                                                        Ω                          ′                                                ;                        s                                                              )                                                  ⁢                                  μ                  ′                                ⁢                                  ⅆ                                      Ω                    ′                                                                                                                                                      ⁢                              Reflected                ⁢                                                                  ⁢                downward                ⁢                                                                  ⁢                flux                                                                        (        8        )            In this equation, μ′ is the cosine of the zenith angle of incoming radiation measured from the ground, fv(Ωs,Ω′;s) is the surface bi-direction reflectance distribution function (BRDF) in units of reciprocal steradians at ground location s, εv(Ωs;s) is the dimensionless directional emissivity in the LOS path direction at s, and the surface reflected downward flux is computed as the upper hemisphere integral (2π sr) over incoming diffuse radiation. The BRDF defines the reflected radiance resulting from surface illumination as a function of both incoming and outgoing angles, and it is normalized to the surface albedo αv(s):
                                          α            v                    ⁡                      (            s            )                          =                              1            π                    ⁢                                    ∫                              2                ⁢                π                                                                                  ⁢                                          ∫                                  2                  ⁢                  π                                                                                              ⁢                                                                    f                    v                                    ⁡                                      (                                          Ω                      ,                                                                        Ω                          ′                                                ;                        s                                                              )                                                  ⁢                                  μ                  ′                                ⁢                                  ⅆ                                      Ω                    ′                                                  ⁢                μ                ⁢                                  ⅆ                  Ω                                                                                        (        9        )            The directional emissivity, the complement of the directional reflectivity ρv(Ωs), dictates the angular distribution of the surface thermal emission, and is computed directly from the BRDF by the following relationship:
                                                        ɛ              v                        ⁡                          (                                                Ω                  s                                ;                s                            )                                ≡                      1            -                                          ρ                v                            ⁡                              (                                                      Ω                    s                                    ;                  s                                )                                                    =                  1          -                                    ∫                              2                ⁢                                                                  ⁢                π                                      ⁢                                                            f                  v                                ⁢                                                                  (                                                      Ω                    s                                    ,                                                            Ω                      ′                                        ;                    s                                                  )                            ⁢                              μ                ′                            ⁢                              ⅆ                                  Ω                  ′                                                                                        (        10        )            
Finally, the Earth viewing radiance is obtained by substituting the surface radiance term, Eq. (8), into the RTE, Eq. (7):
                                                                        I                v                            ⁡                              (                                  Ω                  ;                  0                                )                                      =                                                                                t                    v                                    ⁡                                      (                    s                    )                                                  ⁢                                                      I                    v                                    ⁡                                      (                                          Ω                      ;                      s                                        )                                                              +                                                ∫                                                            t                      v                      abs                                        ⁡                                          (                      s                      )                                                        1                                ⁢                                                                            t                      v                      sct                                        ⁡                                          (                                              s                        ′                                            )                                                        ⁢                                                            B                      v                                        ⁡                                          (                                              T                                                  s                          ′                                                                    )                                                        ⁢                                                                          ⁢                                      ⅆ                                                                  t                        v                        abs                                            ⁡                                              (                                                  s                          ′                                                )                                                                                                        +                                                ∫                                                            t                      v                      sct                                        ⁡                                          (                      s                      )                                                        1                                ⁢                                                                            t                      v                      abs                                        ⁡                                          (                                              s                        ′                                            )                                                        ⁢                                      {                                                                                                                                                                                                                        t                                  v                                  sun                                                                ⁡                                                                  (                                                                      s                                    ′                                                                    )                                                                                            ⁢                                                              F                                v                                sun                                                            ⁢                                                                                                p                                  v                                                                ⁡                                                                  (                                                                      Ω                                    ,                                                                                                                  Ω                                        sun                                                                            ;                                                                              s                                        ′                                                                                                                                              )                                                                                                                      +                                                                                                                                                                                                          ∫                                                              4                                ⁢                                π                                                                                                                                                                                  ⁢                                                                                                                            I                                  v                                                                ⁡                                                                  (                                                                                                            Ω                                      ′                                                                        ;                                    s                                                                    )                                                                                            ⁢                                                                                                p                                  v                                                                ⁡                                                                  (                                                                      Ω                                    ,                                                                                                                  Ω                                        ′                                                                            ;                                                                              s                                        ′                                                                                                                                              )                                                                                            ⁢                                                                                                                          ⁢                                                              ⅆ                                                                  Ω                                  ′                                                                                                                                                                                                          }                                    ⁢                                                                          ⁢                                      ⅆ                                                                  t                        v                        sct                                            ⁡                                              (                                                  s                          ′                                                )                                                                                                                          ;                ⁢                                  ⁢                                            I              v                        ⁡                          (                              Ω                ;                s                            )                                =                                                                                          ɛ                    v                                    ⁡                                      (                                          Ω                      ;                      s                                        )                                                  ⁢                                                      B                    v                                    ⁡                                      (                                          T                      g                                        )                                                              +                                                ∫                                      2                    ⁢                    π                                                                                                          ⁢                                                      [                                                                                                                        t                            v                            sun                                                    ⁡                                                      (                            s                            )                                                                          ⁢                                                  F                          v                          sun                                                ⁢                                                  δ                          ⁡                                                      (                                                          Ω                              -                                                              Ω                                sun                                                                                      )                                                                                              +                                                                        I                          v                                                ⁡                                                  (                                                                                    Ω                              ′                                                        ;                            s                                                    )                                                                                      ]                                    ⁢                                                            f                      v                                        ⁡                                          (                                              Ω                        ,                                                                              Ω                            ′                                                    ;                          s                                                                    )                                                        ⁢                                      μ                    ′                                    ⁢                                      ⅆ                                          Ω                      ′                                                        ⁢                                                                          ⁢                  if                  ⁢                                                                          ⁢                  s                                                      =                                          s                g                            .                                                          (        11        )            In writing this final set of equations, the explicit dependence of the solid angle Ω on position s has been dropped for notational simplicity and the Dirac delta function δ(Ω−Ωsun) has been introduced to move the surface direct solar illumination term inside the angular integration. There is also an implicit understanding that the boundary term Iv(Ω;s) is zero unless the path actually terminates at the ground, s=sg. Eq. (11) serves as the starting point for the derivation of the MODTRAN™ band model.The MODTRAN™ Band Model
The MODTRAN™ band model approach is concerned with calculation of narrow spectral bin radiant intensities <I0(Ω)>:
                              〈                                    I              0                        ⁡                          (              Ω              )                                〉                ≡                              1                          Δ              ⁢                                                          ⁢              v                                ⁢                                    ∫                              Δ                ⁢                                                                  ⁢                v                                      ⁢                                                            I                  v                                ⁡                                  (                                      Ω                    ;                    0                                    )                                            ⁢                                                          ⁢                              ⅆ                v                                                                        (        12        )            The width Δv of the band model spectral bin should be selected sufficiently narrow to insure no more than a modest variation in all spectral data other than molecular absorption. This definition enables all the spectrally smoothly varying properties to be modeled with an effective band value, computed either as an interval average or central value. A smaller band width typically produces finer accuracy. The reformulated band model incorporated into MODTRAN™, described in the parent application which is incorporated herein by reference, provides 4 band model resolution options, 0.1, 1.0, 5.0 and 15.0 cm−1. With an over-line used to represent an effective band value, the band model complement to the LBL LOS radiance expression, Eq. (11), has the following form:
                                                                                          〈                                                            I                      0                                        ⁡                                          (                      Ω                      )                                                        〉                                =                                ⁢                                                      〈                                                                  t                                                  0                          ->                          s                                                                    ⁢                                                                        I                          s                                                ⁡                                                  (                          Ω                          )                                                                                      〉                                    -                                                            ∫                      0                      s                                        ⁢                                                                                            t                                                      0                            ->                                                          s                              ′                                                                                sct                                                _                                            ⁢                                                                        B                          _                                                ⁡                                                  (                                                      T                                                          s                              ′                                                                                )                                                                    ⁢                                                                        ⅆ                                                      [                                                                                                                            t                                                                      0                                    ->                                                                          s                                      ′                                                                                                        part_abs                                                                _                                                            ⁢                                                              〈                                                                  t                                                                      0                                    ->                                                                          s                                      ′                                                                                                        mol_abs                                                                〉                                                                                      ]                                                                                                    ⅆ                                                      s                            ′                                                                                              ⁢                                                                                          ⁢                                              ⅆ                                                  s                          ′                                                                                                      -                                                                                                                        ⁢                                                                            ∫                      0                      s                                        ⁢                                                                                            t                                                      0                            ->                                                                                          s                                ′                                                            ->                              sum                                                                                part_abs                                                _                                            ⁢                                              〈                                                  t                                                      0                            ->                                                                                          s                                ′                                                            ->                              sum                                                                                mol_abs                                                〉                                            ⁢                                                                        t                                                                                    s                              ′                                                        ->                            sum                                                    sct                                                _                                            ⁢                                                                        F                          sun                                                _                                            ⁢                                                                                                    p                                                          s                              ′                                                                                _                                                ⁡                                                  (                                                      Ω                            ,                                                          Ω                              sun                                                                                )                                                                    ⁢                                              (                                                                              ⅆ                                                                                          t                                                                  0                                  ->                                                                      s                                    ′                                                                                                  sct                                                            _                                                                                                            ⅆ                                                          s                              ′                                                                                                      )                                            ⁢                                                                                          ⁢                                              ⅆ                                                  s                          ′                                                                                                      -                                                                                                                        ⁢                                                      ∫                    0                    s                                    ⁢                                                                                    t                                                  0                          ->                                                      s                            ′                                                                          part_abs                                            _                                        ⁢                                          〈                                                                        t                                                      0                            ->                                                          s                              ′                                                                                mol_abs                                                ⁢                                                                              ∫                                                          4                              ⁢                              π                                                                                ⁢                                                                                                                    I                                                                  s                                  ′                                                                                            ⁡                                                              (                                                                  Ω                                  ′                                                                )                                                                                      ⁢                                                                                                                            p                                                                      s                                    ′                                                                                                  _                                                            ⁡                                                              (                                                                  Ω                                  ,                                                                      Ω                                    ′                                                                                                  )                                                                                      ⁢                                                                                                                  ⁢                                                          ⅆ                                                              Ω                                ′                                                                                                                                                        〉                                        ⁢                                          (                                                                        ⅆ                                                                                    t                                                              0                                ->                                                                  s                                  ′                                                                                            sct                                                        _                                                                                                    ⅆ                                                      s                            ′                                                                                              )                                        ⁢                                                                                  ⁢                                          ⅆ                                              s                        ′                                                                                                                                ⁢                                  ⁢        BoundaryCondition        ⁢                                  ⁢                                                                              〈                                                            t                                              0                        ->                        s                                                              ⁢                                                                  I                        s                                            ⁡                                              (                        Ω                        )                                                                              〉                                =                                ⁢                                                                                                                              ɛ                          s                                                _                                            ⁡                                              (                        Ω                        )                                                              ⁢                                                                  t                                                  0                          ->                          s                                                sct                                            _                                        ⁢                                                                  B                        _                                            ⁡                                              (                                                  T                          g                                                )                                                              ⁢                                                                  t                                                  0                          ->                                                      s                            ′                                                                          part_abs                                            _                                        ⁢                                          〈                                              t                                                  0                          ->                          s                                                mol_abs                                            〉                                                        +                                                                                                                        ⁢                                                                                                    t                                                  0                          ->                                                      s                            ->                            sun                                                                          sct                                            _                                        ⁢                                                                  t                                                  0                          ->                                                      s                            ->                            sun                                                                          part_abs                                            _                                        ⁢                                          〈                                              t                                                  0                          ->                                                      s                            ->                            sun                                                                          mol_abs                                            〉                                        ⁢                                                                  F                        sun                                            _                                        ⁢                                                                                            f                          s                                                _                                            ⁡                                              (                                                  Ω                          ,                                                      Ω                            sun                                                                          )                                                              ⁢                                          μ                      sun                                                        +                                                                                                                        ⁢                                                                            t                                              0                        ->                        s                                            sct                                        _                                    ⁢                                                            t                                              0                        ->                        s                                            part_abs                                        _                                    ⁢                                                            〈                                                                        t                                                      0                            ->                            s                                                    mol_abs                                                ⁢                                                                              ∫                                                          2                              ⁢                              π                                                                                ⁢                                                                                                                    I                                s                                                            ⁡                                                              (                                                                  Ω                                  ′                                                                )                                                                                      ⁢                                                                                                                            f                                  s                                                                _                                                            ⁡                                                              (                                                                  Ω                                  ,                                                                      Ω                                    ′                                                                                                  )                                                                                      ⁢                                                                                                                  ⁢                                                          μ                              ′                                                        ⁢                                                          ⅆ                                                              Ω                                ′                                                                                                                                                        〉                                        .                                                                                                          (        13        )            In these equations, the LOS integrands are expressed as functions of the refractive path length s′. Furthermore, the attenuation resulting from absorption is partitioned into its molecular and particulate contributions. The labeling of the band-averaged absorption as arising solely from particulate sources is somewhat misleading. Detailed line compilation data is unavailable for some molecular species, especially large molecules such as the chiorofluorocarbons (CFC's). For these species, absorption is modeled via relatively coarse spectral resolution cross-section data. The resulting transmittances are band averaged values and included in the tpart—abs terms.
Eq. (13) is the fundamental narrow band model radiance equation. It explicitly partitions the spectral data for which effective band values are used from the highly spectrally varying molecular absorption contributions. The effective coarse spectral resolution band data falls into three categories.                1. Ground properties: The surface emissivity εs and the surface BRDF fs;        2. Particulate, Molecular Cross-Section & Rayleigh data: The scattering phase function ps′ and the atmospheric attenuation resulting from scattering tsct and particulate (and molecular cross-section) absorption tpart—abs; and        3. Source functions: The Planck blackbody function B and the TOA solar irradiance Fsun.The ground is comprised of solids and liquids. Employing lower spectral resolution surface data is justified because condensed matter does not exhibit the fine spectral structure of molecular gas absorption. Similarly, even though scattering data can be highly structured for a collection of particles of a specific size and shape; terrestrial aerosols and clouds consist of particulates of various size, shape and orientation. These distributions smooth out the fine spectral structure. Rayleigh scattering optical properties are also spectrally smoothly varying, and the molecular cross-section data is modeled as spectrally smoothly varying in the absence of accurate fine spectral resolution information. The variation of the Planck blackbody function at atmospheric temperatures (˜180 to 305K) for the MODTRAN™ band model widths is modest for the spectral range over which thermal emission is significant.        
The one term for which spectral averaging is somewhat suspect is the TOA solar irradiance. The sun is essentially a ˜6000K blackbody; however, molecular absorption within the solar atmosphere generates Fraunhofer structure. Two facts help support the spectral averaging of the solar data approximation: (1) The spectral correlation between the solar spectrum and the Earth atmospheric absorption is well represented as randomly correlated for most spectral bins; and (2) The current state-of-the-art in measurement and calculation of TOA solar irradiances suggests that spectral structure assignments finer than ˜1 cm−1 are unreliable.
Currently, the multiple scattering (MS) algorithms within MODTRAN™ only support specification of surface reflectance via an angularly independent spectral albedo, fv(Ω,Ω′;s)=αv(s)/π. Given this limitation, the surface reflected downward flux is approximated as a product of the directional reflectivity and the hemisphere integrated downward flux, Fv↓(s):
                                          ∫                          2              ⁢                                                          ⁢              π                                ⁢                                                    I                                  v                  ⁢                                                                                                    ⁡                              (                                                      Ω                    ′                                    ;                  s                                )                                      ⁢                                          f                v                            ⁡                              (                                  Ω                  ,                                                            Ω                      ′                                        ;                    s                                                  )                                      ⁢                          μ              ′                        ⁢                          ⅆ                              Ω                ′                                                    ≅                                            ρ                              v                ⁢                                                                                        ⁡                          (                              Ω                ;                s                            )                                ⁢                                    F              v              ↓                        ⁡                          (              s              )                                      ≡                                            ρ              v                        ⁡                          (                              Ω                ;                s                            )                                ⁢                                    ∫                              2                ⁢                                                                  ⁢                π                                      ⁢                                                            I                  v                                ⁢                                                                  (                                                      Ω                    ′                                    ;                  s                                )                            ⁢                              μ                ′                            ⁢                                                ⅆ                                      Ω                    ′                                                  .                                                                        (        14        )            This expression for the reflected downward flux is quite reasonable if the distribution of downward diffuse radiance is nearly homogeneous or the surface reflectance nearly Lambertian. On the other hand, for clear sky solar illumination of a specular surface such as still water, Eq. (14) is a poor approximation; however, in this case, the observed radiance will be dominated by the direct solar illumination and the error in total observed radiance may still be small. Merging the flux approximation into the narrow band model radiance expression, Eq. (13), the surface reflected diffuse radiance term factors as follows:
                              〈                                    t                                                                                ⁢                                  0                  ⁢                                                                          ->                                                                          ⁢                  s                                                                                                              ⁢                mol_abs                                      ⁢                                          ∫                                  2                  ⁢                  π                                            ⁢                                                                    I                    s                                    ⁡                                      (                                          Ω                      ′                                        )                                                  ⁢                                                                            f                      s                                        _                                    ⁡                                      (                                          Ω                      ,                                              Ω                        ′                                                              )                                                  ⁢                                  μ                  ′                                ⁢                                                                  ⁢                                  ⅆ                                      Ω                    ′                                                                                〉                ≅                                                            ρ                s                            _                        ⁡                          (              Ω              )                                ⁢                                    〈                                                t                                                                                                    ⁢                                          0                      ⁢                                                                                          ->                                                                                          ⁢                      s                                                                                                                                          ⁢                    mol_abs                                                  ⁢                                                      F                    v                    ↓                                    ⁡                                      (                    s                    )                                                              〉                        .                                              (        15        )            
When MODTRAN™ utilizes its correlated-k (CK) option, an algorithm that recasts the molecular band model into weighted sum of monochromatic calculations, the path radiant intensity computation follows the prescription of Eq. (13) with the caveat of Eq. (15). However, further approximations are necessary if the band model algorithm is used without CK. The issue here is that the MODTRAN™ atmospherically scattered diffuse radiance and the downward surface flux terms are computed from the MS algorithms within MODTRAN™. These MS algorithms were developed to solve the coupled monochromatic radiation transport problem, and they rely heavily upon Beer's Law. Beer's Law states that the spectral transmittance of a multi-segmented path equals the product of the individual segment transmittances:tv(0→s′→s)=tv(0→s′) tv(s′→s).  (16)Alternatively stated, segment extinction optical depths are additive. This relationship does hold for CK transmittances but not for band model in-band transmittances:<t0→s′→s>≠<t0→s><ts′→s>.  (17)When the CK option is not invoked, MODTRAN™ proceeds with multiple scatter source term and level flux calculations computed from a Beer's Law model even though the layer optical depths are poorly defined. The inaccuracy of this approach is limited somewhat by combining the MS source terms with foreground transmittances computed for the inhomogeneous path. Thus, the LOS radiance terms containing diffuse radiation are calculated as follows:
                                              ⁢                                                                                                  〈                                                                  t                                                  0                          ->                          s                                                mol_abs                                            ⁢                                              F                        s                        ↓                                                              〉                                    ≅                                                            〈                                              t                                                  0                          ->                          s                                                mol_abs                                            〉                                        ⁢                                          〈                                              F                        s                        ↓                                            〉                                                                                                                    band                  ⁢                                                                          ⁢                  model                                                                                                                          〈                                                                  t                                                  0                          ->                                                      s                            ′                                                                          mol_abs                                            ⁢                                                                        ∫                                                      4                            ⁢                                                                                                                  ⁢                            π                                                                          ⁢                                                                                                            I                                                              s                                ′                                                                                      ⁢                                                                                                                  (                                                          Ω                              ′                                                        )                                                    ⁢                                                                                                                    p                                                                  s                                  ′                                                                                            _                                                        ⁡                                                          (                                                              Ω                                ,                                                                  Ω                                  ′                                                                                            )                                                                                ⁢                                                      ⅆ                                                          Ω                              ′                                                                                                                                            〉                                    ≅                                                            〈                                              t                                                  0                          ->                          s                                                mol_abs                                            〉                                        ⁢                                                                  ∫                                                  4                          ⁢                                                                                                          ⁢                          π                                                                    ⁢                                                                        〈                                                                                    I                                                              s                                ′                                                                                      ⁢                                                                                                                  (                                                          Ω                              ′                                                        )                                                    〉                                                ⁢                                                                                                            p                                                              s                                ′                                                                                      _                                                    ⁡                                                      (                                                          Ω                              ,                                                              Ω                                ′                                                                                      )                                                                          ⁢                                                  ⅆ                                                      Ω                            ′                                                                                                                                                                                            with                  ⁢                                                                          ⁢                  no                  ⁢                                                                                                    ⁢                                                                                                  ⁢                  CK                                                              .                                    (        18        )            
Together, Eqs. (13), (15) and (18) define the MODTRAN™ path radiance formulation. However, implementation of the approach still requires a description of three key model components: the MODTRAN™ statistical band model molecular transmittance calculation, the multiple scattering algorithms, and the surface water model. Descriptions of these components follow.